A New Power Flow Model Incorporating Effects of ... - CiteSeerX

WSEAS TRANSACTIONS on POWER SYSTEMS Manuscript received June 30, 2007; revised Aug. 4, 2007

Jagabondhu Hazra and Avinash Kumar Sinha

A New Power Flow Model Incorporating Effects of Automatic Controllers JAGABONDHU HAZRA AND AVINASH KUMAR SINHA Department of Electrical Engineering Indian Institute of Technology, Kharagpur Kharagpur, West-Bengal-721302 INDIA [email protected], [email protected] Abstract: - This paper presents a modified fast decoupled load flow method which incorporates voltage and frequency dependent characteristics of loads, generator’s regulation characteristics and sensitivity based automatic transformer tap adjustment scheme. The proposed method is suitable for analysis of large power systems in the normal as well as abnormal conditions such as major contingencies involving line outage, generator and/or load loss, system islanding etc. The proposed method is tested on IEEE test systems and an Indian system. Simulation results are presented to show the characteristics of the proposed method.

Key-Words: - Load flow, load characteristics, regulation characteristics, tap adjustment voltage and frequency dependent characteristics of load, generator regulation characteristics and effect of automatic controllers such as transformer on load tap adjustment. The proposed method can carryout load flow study even if the system gets islanded due to line outages. Results comparing the efficiency of the proposed method with the existing load flow methods are presented.

1 Introduction Load flow is one of the most important studies in power system operation and planning. Well known Newton-Raphson Load Flow (NRLF) method gives very accurate results and converges in most of the cases [1]. But formation of jacobian matrix in each iteration makes this method time consuming. Large computation time by NRLF was the motivation behind development of Fast Decoupled Load Flow (FDLF) method [2]. FDLF is the most commonly used load flow method because of its simplicity and faster solution. To improve the performance of FDLF method several modifications have been suggested in [3-4].Recently some AI techniques [56] has also used to solve the power flow problem. Conventional power flow methods assume loads as constant (independent of voltage and frequency variation) and supply demand balance is regulated by the slack bus generation. These assumptions may be valid during normal operation of power system but when contingencies like major generator outage or tie line outage occur, system frequency and voltages may deviate considerably from the normal. Load flow methods introducing load and generator characteristics and effects of system control devices have been proposed in [7-8]. These methods require significant amount of time for the solution as the methods are based on NewtonRaphson solution technique requiring calculation of the jacobian matrix in each iteration. In this paper, a Modified Fast Decoupled Load Flow method is proposed which considers the ISSN: 1790-5060

2 Load Flow Model 2.1 Load characteristics Both real and reactive power loads are assumed to be composed of three parts. One part is constant independent of change in voltage, another part is proportional to the voltage and third part is constant impedence load which changes as a square of the voltage. Loads are also assumed to change with change in frequency. Frequency and voltage dependent loads can be expressed as: (1) Pdi = Pdoi (1 + ki ∆f )(ai + biVi + ciVi 2 ) 2 (2) Qdi = Qdoi (1 + ki′∆f )(ai′ + bi′Vi + ci′Vi ) where, Pdi , Qdi Actual active and reactive load at bus i Pdoi , Qdoi Active and reactive load at bus i for nominal voltage and nominal frequency System frequency deviation ∆f Bus voltage magnitude at bus i Vi

202

Issue 8, Volume 2, August 2007

WSEAS TRANSACTIONS on POWER SYSTEMS


n  =  ∑ Vi V j Yij sin(θij + δ j − δ i )  ∆δ i  j =1   j ≠i 

ki , ai , bi , ci Coefficients of active load at bus i with ai + bi + ci = 1 ki′, ai′, bi′, ci′ Coefficients of reactive load at bus i with ai′ + bi′ + ci′ = 1 All quantities are in per unit value.

n

−∑  Vi V j Yij sin(θ ij + δ j − δ i )∆δ j  j =1 j ≠i

2   ngi P +  ∑ max ik + Pdoi × ki × ai + bi Vi + ci Vi  ∆(∆f ) (8) R  k =1 ik 

(

2.2 Generator characteristics

2 =  −Qi − Bii Vi  ∆δ i  

Real power output of generators with free governor operation is adjusted by the governor droop characteristics. This can be expressed as: ngi

ngi

k =1

k =1

Pgi = ∑ Pgik = ∑ ( Psetik −

Pmax ik ∆f ) Rik

Pmin ik ≤ Pgik ≤ Pmax ik

n

−∑  Vi V j ( Bij cos δ ij − Gij sin δ ij ) ∆δ j  j =1 j ≠i

 ngi P 2  +  ∑ max ik + Pdoi × ki × ai + bi Vi + ci Vi  ∆ (∆f ) (9) R  k =1 ik 

(3)

(

(4)

2

Number of generators at bus i Total active power generation at bus i Active power generation of kth unit at bus i

Pgi Pgik

  ngi n P ∆Pi =  − Bii × ∆δ i − ∑  Bij × ∆δ j   Vi + ∑ max ik ∆(∆f )   j =1 k =1 Rik j ≠i  

Psetik Set value of active power generation of kth

(

+ Pdoi × ki × ai + bi Vi + ci Vi

unit at bus i Pmax ik , Pmin ik maximum and minimum generations of kth generator at bus i th Rik Regulation constant of k generator at bus i. All quantities are in per unit value.

n

= [ − Bii ∆δ i − ∑ Bij ∆δ j ] Vi j =1 j ≠i

 ngi P  +  ∑ max ik ∆(∆f ) + bi × ki × Pdoi ∆(∆f )  Vi  k =1 Rik × Vi  n

(5)

j =1 j ≠i

 ngi P  +  ∑ max ik + bi ki Pdoi  ∆(∆f )  k =1 R × V  ik i  

Active power mismatch at bus i can be written as: ∆Pi = Pi , cal − Pi , sch = Pi , cal − ( Pgi − Pdi )

2

= ∑ Vi V j Yij cos(θ ij + δ j − δ i ) j =1

Without loss of generality, bus n is assumed as reference bus with δ n = 0 , therefore above equation in matrix form can be represented as follows:

ngi

−∑ ( Psetik − ∆f × Pmax ik / Rik ) 2

)

 ∆P1′    V   − B11  1    ∆P2′      − B21  V2  =   .   .     ∆Pn′    V   − Bn1  n  

Pmax1k  + b1k1 Pdo1  k =1 1k V1  ∆δ 1  ng 2  Pmax 2 k − B22 . ∑ + b2 k2 Pdo 2   ∆δ 2    k =1 R2 k V2  .  . . .   ∆ (∆f )  ngn  Pmax nk − Bn 2 . ∑ + bn k n Pdon  k =1 Rnk Vn   ∆δ  ′ ] (13) In general, ∆P ′ / V  = [ Bmod   ∆(∆f ) 

(6)

Change in bus voltage angles and frequency is predominantly influenced by the change in real power injections. Therefore change in real power injection at bus i can be written as: n ∂Pi ∂P ∂P ∆δ i + ∑ i ∆δ j + i ∆ (∆f ) δ ∂δ i ∂ ∂∆ f j =1 j

(7)

j ≠i

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(12)

where, ∆Pi ′ = ∆Pi − Pdoi × ki × (ai + ci Vi )∆(∆f )

n

∆Pi =

(11)

∆ Pi ′ Vi = − Bii ∆δ i − ∑ Bij ∆δ j

j =1

(

(10)

2

n

+ Pdoi (1 + ki ∆f ) ai + bi Vi + ci Vi

) ∆(∆f )

∆Pi − Pdoi × ki × (ai + ci Vi )∆(∆f )

Static load flow equations for real power is given by

k =1

2

Rearranging,

2.3 Real power load flow equations Pi ,cal = ∑ Vi V j Yij cos(θ ij + δ j − δ i ) i ∀ all buses

)

Making the assumptions Bii Vi >> Qi , cos δ ij ≅ 1 and Bij >> Gij Sinδ ij

where, ngi

)

203

ng 1

− B12

.

∑R




′ ] , Bij terms are constant and same In matrix [ Bmod

n

or, ∆Qi′ / Vi = [ − Bii + 2ci′Qdoi ] ∆Vi − ∑  Bij ∆V j 

as Bij′ terms in Fast Decoupled Load Flow [2]. The

where, ∆Qi′ = ∆Qi − Qdoi bi′ + ki′∆f (bi′ + 2ci′ Vi )  ∆Vi

terms bi ki Pdoi , Pmax ik , Rik and generator bus voltage (PV bus) Vi are also constant. For load buses voltage is not constant but as there is no generation,

Above equation in matrix form can be represented as follows:

P Pmax ik term is zero hence max ik term becomes zero. Rik Vi

 ∆Q1′ V1   − B11 + 2c1′Qdo1  ′   − B21  ∆Q2 V2  =     . .    ′ ∆ − Q V B  n n   n1

′ ] matrix is a constant matrix. Therefore, [ Bmod 2.4 Reactive power load flow equations

− B12

− B1n   ∆V1    ∆V  . − B2 n  2  .  . .   . − Bnn + 2cn′ Qdon   ∆Vn  .

− B22 + 2c2′ Qdo 2 . − Bn 2

′′ ][ ∆V ] In general, ∆  Q ′ / V  = [ Bmod

Static load flow equations for reactive power for any load bus i is given by:

(20)

′′ ] As Bij and ci′Qdoi terms are constant, matrix [ Bmod

n

Qi ,cal = −∑ Vi V j Yij sin(θ ij + δ j − δ i ) ;

(19)

j =1 j ≠i

(14)

is also a constant matrix.

j =1

Reactive power mismatch at load bus i is given by: ∆Qi = Qi ,cal − Qi , sch = Qi − (Qgi − Qdi )

3 Transformer tap adjustment

n

= −∑ Vi V j Yij sin(θij + δ j − δ i )

Equivalent network model of any voltage regulating transformer connected between buses i and j with tap on bus i can be represented as shown in Fig. 1.

j =1

(

+Qdoi (1 + ki′∆f ) ai′ + bi′ Vi + ci′ Vi

2

)

(15)

Change in bus voltage magnitudes is predominantly influenced by the change in reactive power injections. Therefore change in reactive power injection at bus bars can be written as: ∆Qi =

Vj

Vi ytij (tij − 1)

n ∂Qi ∂Q ∆Vi + ∑ i ∆V j ∂Vi j =1 ∂V j

ytij

(1 − tij ) y

Fig.1 : Representation of voltage regulating transformer

Reactive power flow through the transformer from bus i to bus j can be written as:

j ≠i

 n  =  −∑ V j Yij sin(θ ij + δ j − δ i ) − 2 Vi Bii  ∆Vi  j =1    j ≠ i +Qdoi (1 + ki′∆f )(bi′ + 2ci′ Vi )∆Vi

Qij = − Im Vi ∗ {Vi × ytij (tij − 1) + (Vi − V j )tij y}

= − Im Vi 2 tij2 y − Vi ∗V j tij y  = −Vi 2 tij2 b − Vi V j y tij sin(δ i − δ j − θij )

n

+ ∑  − Vi Yij sin(θ ij + δ j − δ i )  ∆V j

Similarly, Q ji = −V j2 b − Vi V j y tij sin(δ j − δ i − θ ij ) (22)

(16)

j =1 j ≠i

Where, y (= g + jb) is the admittance of the transformer; Vi , V j are complex voltages at bus i and

=  (Qi + Bii Vi ) / Vi − 2 Vi Bii + Qdoi (1 + ki′∆f )(bi′ + 2ci′ Vi )  ∆Vi   2

n

−∑ Vi ( Bij cos δ ij − Gij sin δ ij ) ∆V j

j respectively; Vi* is complex conjugate of Vi ; tij is tap position. Q-V load flow equation given by equation (20):

(17)

j =1 j ≠i

Now, BiiVi 2 >> Qi , cos δ ij ≅ 1 and Bij >> Gij Sinδ ij Therefore,

 ∆Q′  ′′ ][ ∆V ]   = [ Bmod  V 

∆Qi = − Bii Vi + Qdoi (1 + ki′∆f )(bi′ + 2ci′ Vi )  ∆Vi n

−∑ Bij Vi ∆V j

 ∆Q ′   = [X ]  V 

′′ ]  or, [ ∆V ] = [ Bmod −1

(18)

j =1 j ≠i

 ∆Q ′     V 

(23)

Now for ith bus this can be written as:

Rearranging,

∆Vi = X i1 × ∆Q1′ / V1 + X i 2 × ∆Q2′ / V2 + ...... + X in × ∆Qn′ / Vn

∆Qi − Qdoi bi′ + ki′∆f (bi′ + 2ci′ Vi )  ∆Vi

Change in bus voltage with respect to change in transformer tap tij between bus i and j

n

= − Bii Vi ∆Vi + 2ci′Qdoi Vi ∆Vi − ∑  Bij Vi ∆V j  j =1 j ≠i

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(21)

204



∆Vi = X ii ×


1 ∂Qi 1 ∂Q j × ∆tij + X ij × × ∆tij Vi ∂tij V j ∂tij

Sensitivity, Si ≡

buses,

(24)

and

∆Vi 1 ∂Qi 1 = X ii × × + X ij × × Vi ∂tij ∆tij V j ∂tij

∂Qi = −2Vi 2 tij b − Vi V j y sin(δ i − δ j − θ ij ) ∂tij ∂Q j ∂tij

′′ ] [ Bmod

diagonal

elements

Bij = B ji = 10−6 (a very small number) and subtracting

= − Vi V j y sin(δ j − δ i − θ ij )

Bij from Bii and B jj . Similar modifications are made

After convergence of load flow sensitivity for each transformer is calculated using (25) if regulated voltage is not within the limits and then based on sensitivity factors taps are adjusted to maintain desired voltages. Normally, one iteration of load flow is required to get the solution after tap adjustment.

in

′′ ] [ Bmod

matrix also. This does not affect the

accuracy of the method as indicated by the simulation results, but considerably improves the computation effort and time. Proposed load flow method can carry out multi area load flow study when the system gets islanded due to line tripping. In case system gets islanded, it is required to calculate [ B′] and [ B′′] matrices for

4 Solution strategy

each island and for such case storage, ordering and reduction of the sparse matrices are required for each matrix separately which takes reasonable time. ′ ] But in the proposed method same matrices [ Bmod

In this paper sequential AC-DC iterative solution method is used to conduct load flow for ac/dc system [9]. In the proposed method no slack bus is required because load generation balance is regulated by all the generators according to their regulation characteristics. One bus (say bus n) is considered as reference bus and all other bus voltage angles are computed with respect to the reference. AC load flow solutions are obtained by iterative solution of (13) and (20). Bi-Factorization technique [10] is used to solve equations (13) and (20). As both the matrices in (13) and (20) are constant those matrices need to be formed and factorized only once ′′ ] in (20) during the load flow solution. Matrix [ Bmod

′′ ] of the integrated system are used with and [ Bmod small modifications (see Appendix for details).

5 Simulation results and discussions In order to verify the effectiveness of the proposed load flow method it is compared with other methods ([2], [7] and [11]) on IEEE 24 bus reliability test system, IEEE 118 bus system and NREB (Northern Region Electricity Board, India) 390 bus system having 754 transmission lines, 205 transformers, 393 generators, one SVS and one HVDC bipolar link. For simulation purpose regulation parameter (Ri) is taken as 5 percent for all the generators. Coefficients of active and reactive load for all the loads are taken as follows:

is a constant sparse matrix with symmetric structure ′ ] in (13) is a constant spares whereas matrix [ Bmod matrix but its structure is asymmetric. To make ′ ] structurally symmetric a very small matrix [ Bmod value (10-6) is put in every ij position if value at ij position is zero but value at ji position is non zero and vice versa. In FDLF [ B′] matrix does not include row and

a = .85 b = .10 c = .05 k = 1.5 a′ = .80 b′ = .15 c′ = .05 k ′ = 1.5

column corresponding to slack bus and [ B′′] matrix

Table 1 illustrates the comparison of computational times for the different methods. For all the methods same sparse matrix solution technique is used and convergence limit is taken as 10-4 per unit. The simulation is carried out on a personal computer with Pentium IV, 3.0 GHz processor, 512 MB RAM and Windows-XP operating system. From Table 1, it can be seen that

does not include rows and columns corresponding to slack bus as well as PV buses. In the proposed ′ ] and [ Bmod ′′ ] includes method, each matrix [ Bmod rows and columns corresponding to all the buses in the systems (i.e. of dimension n × n for n bus system). To take care of constant voltages at PV

ISSN: 1790-5060

matrix

corresponding to PV buses are replaced by a large value (106). The advantage of using the size of each matrix as n × n is that in case of line outage contingency analysis same ordered matrices of the integrated system can be directly used which saves reasonable computational time. For outage of line ij, ′ ] matrix is modified by placing [ Bmod

∂Q j

(25) where,

in

205




the execution time of the proposed method is slightly higher than the conventional FDLF method and it is much less compared to the NRLF and the method in [7]. Proposed method takes a little more time to converge than the FDLF because of extra ′ ] , [ Bmod ′′ ] and calculation while forming [ Bmod

takes 43.1 (27.09+16.01) ms. The two methods provide same results (within the tolerance (10-4) accuracy). This shows that the proposed method provides accurate results for multi-area power flow at one go whereas the method in [7] and NRLF methods require separate solutions for each island. Performance of the proposed method is also tested for major contingencies like outage of large generator unit. Simulation results for outage of 607 MW generation at bus 89 for the IEEE 118 bus test system are presented in Table 4. Post contingency steady state generations of all the generators are calculated using the proposed method, FDLF method and using the method in [11]. FDLF method assigns all the imbalance between load and generation to the slack bus, method in [11] assigns the imbalance among all the generators according to their inertia (may be valid immediately after the contingency but not in steady state after the contingency) whereas in the proposed method, imbalance is adjusted at all other generators according to their regulation characteristics which results in deviation of system frequency, this is what will happen in actual system during operation. In Table 4 post contingency line flows using different

correction vectors to take into account load and generation characteristics which are not accounted in FDLF method. Test results showing the effectiveness of automatic tap adjustment scheme are presented in Table 2. Results show that transformer taps are automatically adjusted within one iteration by the proposed LF method to maintain the specified voltages. As taps are changed in discrete steps (.0125 pu/step) therefore it may not be possible to get exact desired voltage at the controlled bus. The proposed power flow method is capable of performing the power flow at the same time for all the islands when the system gets islanded. Conventional FDLF method and method in [7] can not handle the multi area power flow because of convergence problem caused by islanding of the system. To show the effectiveness of the proposed method for conducting multi area power flow, simulation is carried out on IEEE 118 bus system. To create two islands for the 118 bus system 7 transmission lines between buses 19-20, 17-31, 6568, 69-47, 69-49, 17-113 and 25-26 are tripped. To compare the accuracy of the proposed method in case of system islanding, power flow is also carried out using the method in [7] for each island separately. Table 3 shows that the proposed method takes only 4.5 ms to conduct the power flow where as method in [7]

Table 3: Multi area power flow study

TB

Vsh

30* 38* 63* 64*

17 37 59 61

1.00 1.00 1.00 1.00

Without control V(p.u.) Tap .985 .960 .961 .935 .968 .960 .983 .985

Tap .9725 1.01 1.02 .9975

* indicates tap side; FB from bus; TB to bus; Vsh schedule voltage; V actual voltage.

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Exe.Time (ms)

Freq. (Hz)

4.5

49.999

Voltage Angel (radian) .3105 .3195 .3259 .3907 .3985 .4753 .5236 .3231 .2981 .1887

49.998

Voltage magnitude

Exe Time (ms) 27.09 16.01

Freq.(Hz)

FDLF

With control V(p.u.) 1.001 1.002 1.002 1.001

.9550 .9714 .9677 .9980 1.002 1.002 1.035 .9840 .9861 .9800

Table 4: Line flows for outage of a generator unit at bus 89 for the IEEE 118 bus system

Table 2: Results for the tap adjustment scheme FB

49.999

(ms) 1.05 4.16 23.41

Proposed method

49.998

(ms) 0.55 2.95 15.3

Voltage Angel (radian)

(ms) 7.46 16.49 162.3

.3106 .3195 .3260 .3907 .3986 .4753 .5236 .3232 .2982 .1888

206

Cap

Proposed

.9550 .9714 .9677 .9980 1.002 1.002 1.035 .9840 .9861 .9800

TB

FDLF

Voltage magnitude

NRLF

1 1 1 1 1 2 2 2 2 2

FB

24 Bus 118 Bus NREB

Method in [7] (ms) 8.25 33.49 248.5

1 2 3 4 5 68 69 70 71 72

Line

Test System

Island

Table 1: Comparison of execution time

Bus

Method in [7]

3 34 90 97 100 111 186

4 8 38 65 69 69 81

5 30 65 68 70 77 80

100 100 100 100 100 200 300

MVA Flow 107 79 183 60 160 304 311

Method in [11] MVA Flow 105 78 182 75 162 308 291

Proposed method MVA Flow 85 145 97 295 83 191 260




[6] Adnan S. Borisly and A. K. Al-Othman, Solution of Load-Flow Problem using Fuzzy Linear Regression Approach, 2nd WSEAS Int. Conf. on CIRCUITS, SYSTEMS, SIGNAL and TELECOMMUNICATIONS, 2008, pp. 17-22. [7] M. Okamura, Y. Oura, S. Hayashi, K. Uemura and F. Ishiguro, A new power flow model and solution method- Introducing load and generator characteristics and effects of system control devices’, IEEE Trans. Power Apparatus and Systems, Vol. PAS-94, No. 3, 1975, pp. 1042-1050. [8] M. S. Calovic and V. C. Strezoski, Calculation of steady state load flows incorporating system control effects and consumer self-regulation characteristics, Int J Electr Power Energy Syst., Vol 3, No 2, 1981, pp. 65-74. [9] R. S. Kuruneru, A. Bose, and R. Bunch, Modeling of High Voltage Direct Current Transmission Systems for Operator Training Simulators, IEEE Trans. Power Systems, Vol. 9, No. 2, 1994, pp. 714-720. [10] K. Zollenkopf, Bi-factorization-Basic computational algorithm and programming techniques, in Large Sparse Sets of Linear Equations, J. K. Ried, Ed. New York: Academic Press, 1971, pp. 75-96. [11] D. Hazarika, S Bhuyan, and S. P. Chowdhury, Line outage contingency analysis including the system islanding scenario, Int J Electr Power Energy Syst, vol. 28, 2006, pp. 232-243.

methods are presented. From the results in Table 4 it is clear that some lines which are over loaded according to FDLF method and method in [11] are not overload according to the proposed method and vice versa. FDLF method and method in [11] give misleading result because of their method of distribution of the generation loss and moreover those methods do not consider the effect of automatic controllers for major contingencies. After tripping of the generator, system frequency drops down from 50.00 Hz to 49.90 Hz. As system frequency drops total system load reduces from 4229 MW to 4215 MW. Conventional methods do not consider these changes.

6 Conclusions This paper proposes a new power flow algorithm that incorporates the effect of frequency and voltage dependent load, generators regulation characteristics and sensitivity based on-load automatic tap adjustment scheme. The method is computationally efficient and can perform multi-area power flow study accurately. The proposed method may be useful for power system planning and operation including the study of load shedding, generator shedding, system islanding, emergency controls etc. for power system security analysis. In future, FACTS devices models will be incorporated in the proposed method. References: [1] B. Stott, Review of Load Flow Calculation Methods, IEEE Proc., Vol. 62, No. 7, 1974, pp. 916-929. [2] B. Stott, and O. Alsac, Fast Decoupled Load Flow, IEEE Trans. Power Apparatus and Systems, Vol. PAS-93, 1974, pp. 859-869. [3] Paul H. Haley, and Mark Ayres, Super decoupled loadflow with distributed slack bus, IEEE Trans. Power Apparatus and Systems, Vol. PAS-104, No. 1, 1985, pp. 104-113. [4] D. Rajicic and A. Bose, A Modification to the Fast Decoupled Power Flow for Networks with High R/X Ratios, IEEE Trans. Power Systems, Vol. 3, No. 2, 1988, pp. 743-746. [5] F. Riganti Fulginei and A. Salvini, Bacterial Chemotaxis Algorithm for Load Flow Optimization, Proc. of the 5th WSEAS/IASME Int. Conf. on Electric Power Systems, High Voltages, Electric Machines, 2005, pp. 436438.

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Appendix

3 Island 2

1

Island1 2

Fig.2: Islanded system

Line 2-3 is tripped to make 2 islands. Modified matrices after islanding are as follows:  ng1 Pmax1k  ∑ R V + +b1k1 Pdo1  k =1 1k 1 ′ = Bmod b2 k2 Pdo 2   0    106 − B12  ′′ =  − B21 − B22 ′ + 2c2′ Qd 02 Bmod  0 10−6 

207

− B12 ′ − B22 10−6

0   10−6  6  10 

   −6  10  ng 3 Pmax 3k + +b3k3 Pdo 3  ∑ R V k =1 3 k 3 

0

; B22′ = B22 − B23
